Although it doesn't actually appear in the form of an angel, Rincewind has an argument with his conscience in Sourcery.
Carcer, the villain of Night Watch is described as having a devil on each shoulder, working together and egging him on.
The Omnian priest The Quite Reverend Mightily-Praiseworthy-Are-They-Who-Exalteth-Om Oats has two voices in his head which he thinks of as the Good Oats and the Bad Oats. That is: one of them thinks of itself as the Good Oats who encourages living properly and devotion to Om and the other as the Bad Oats who gives him impious and improper thoughts; but the other one thinks it's the Good Oats who encourages being sensible and thinking logically, and the other one's the Bad Oats that just wants blind obedience to the church. In the end, he's able to get them to work together at last.
The witch Agnes/Perdita Nitt has the same issue, although in this case it's between the kind, clear-headed Agnes and the sassy, dare-devil Perdita.
In Terry Pratchett and Neil Gaiman's Good Omens, the demon Crowley and the angel Azaraphale, detailed by Heaven and Hell respectively to remain in the human world since the expulsion from Eden and, er, guide its steps, could fairly be described as the Angel and the Devil standing on opposite shoulders of the whole human race. This does not work out as their superiors intend: most of the time they informally agree to cancel each other out, spare thenselves the effort, and let Humanity find its own middle way. But just sometimes...
Referenced in John D. Fitzgerald's Great Brain series: the Great Brain's father comments that he must be deaf in the right ear, since that's the one the good angel is supposed to whisper into.
Phaedrus describes the psyche as composed of a Power Trio: a light horse, proud, noble, and good-intentioned; a dark horse, crippled and malevolent; and the charioteer who must keep rein on both. All Psychology Is Freudian, but Freud is, apparently, a copycat.
Plato tells us that he said to have a personal daemon (who was basically a Jimminy Cricket figure). Plato is one of the first recorded instances of people using daemon/daimon to mean something personal and guiding. Earlier it had been used to mean the same thing as deity and only much later was demon exclusively evil. Post-Alexander The Great they started to come up with "Good Demon, Bad Demon".
Deconstructed in The Screwtape Letters, where the title character is writing instructional letters to Wormwood, who is a literal version of the shoulder devil. The Tempters are highly organized, with a training college, middle management, and harsh punishments for devils who fail to corrupt their "patients." At one point, Screwtape orders Wormwood to subvert this trope by impersonating the good angel and arguing with the other angel over who actually is the one giving good advice.
Another C. S. Lewis book and a rare deadly-serious example is Perelandra. The protagonist, Dr. Ransom, (reluctantly) acts as the "good angel" for the Lady of Perelandra (a next-generation Eve) while she is tempted to disobedience by a/the devil (the "bad angel") possessing the body of Ransom's former colleague.
Referenced in Paper Towns when Quentin is blackmailing a classmate into paying for the bikes his friends wrecked.
"I understand that you do not control Chuck and Jasper. But you see, I am in a similar situation. I do not control the little devil sitting on my left shoulder. The devil is saying, 'PRINT THE PICTURE PRINT THE PICTURE TAPE IT UP ALL OVER SCHOOL DO IT DO IT DO IT.' And then on my right shoulder there is a little tiny white angel. And the angel is saying, 'Man, I sure as shit hope all those freshmen get their money bright and early on Monday morning.' So do I, little angel. So do I."
In Andrei Belianin's My Wife Is A Witch duology, the main character, a poet, finds out that his librarian wife is actually descended from a long line of witches. When she disappears into the magical realms because of him, he goes to find her. Once he is exposed to magic, though, he begins to see two figures, both looking like him: an angel in a white robe and with wings and halo named Ancipher and a red, horned (possibly Jewish) demon named Pharmason. Unlike the other cases, these two are real, although no one else can see or hear them. They become the poet's companions (and friends) as he travels through the magical realms looking for his wife and, in the sequel, for her cousin. As can be expected, the angel and demon constantly engage in arguments, usually started by the demon. Ironically, it's the angel who turns these into fist fights. While it may seem that the demon is there only to lead the poet down the wrong path, it's really just his job, and he's actually not a bad guy. In fact, he argues with the poet that if he only had an angel, he'd be married to a nice, proper Christian woman (not a witch), go to church every week, and only have sex for procreation. The demon provides a healthy counterbalance. They can appear either as tiny people on his shoulders or human-sized versions. The second book reveals that Heaven is operated by a Celestial Bureaucracy. Hell then decides to adopt this system, forcing Pharmason to fill out and submit multiple reports in triplicate every day.
In The Adventures of Huckleberry Finn, when Jim is telling Huck's dad's fortune, he says that there are two angels at his side: a good, white one at his right shoulder, and an evil, black one at his left.
This appears late in William Shakespeare's Sonnets and is interesting in that his Devil seduces the Angel. While it's not literally presented as them being on his shoulder, and the Angel and Devil represent the Young Man and the Dark Lady, respectively, it is carried out in much the same way as a classical Psychomachia.